I'm also looking to replicate that switch for "Enable offline mode" via ArcGIS Python API, but am not having much luck figuring out how. Did you ever figure this out? All I can find are options for managing actual offline areas, which I do not want to do.
Additionally, I'm also wanting to change the "Offline: Advanced Options". I found this documentation, but it doesn't seem to be working for me. When I submit a dictionary key and value, as shown below, all I get is "TypeError: 'dict' object is not callable". Am I doing something wrong?
webmap_obj.offline_areas.offline_properties({"download":"features"})